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Simplifying 3x * 3x + -24x + 40 = 100 Reorder the terms for easier multiplication: 3 * 3x * x + -24x + 40 = 100 Multiply 3 * 3 9x * x + -24x + 40 = 100 Multiply x * x 9x2 + -24x + 40 = 100 Reorder the terms: 40 + -24x + 9x2 = 100 Solving 40 + -24x + 9x2 = 100 Solving for variable 'x'. Reorder the terms: 40 + -100 + -24x + 9x2 = 100 + -100 Combine like terms: 40 + -100 = -60 -60 + -24x + 9x2 = 100 + -100 Combine like terms: 100 + -100 = 0 -60 + -24x + 9x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-20 + -8x + 3x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-20 + -8x + 3x2)' equal to zero and attempt to solve: Simplifying -20 + -8x + 3x2 = 0 Solving -20 + -8x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6.666666667 + -2.666666667x + x2 = 0 Move the constant term to the right: Add '6.666666667' to each side of the equation. -6.666666667 + -2.666666667x + 6.666666667 + x2 = 0 + 6.666666667 Reorder the terms: -6.666666667 + 6.666666667 + -2.666666667x + x2 = 0 + 6.666666667 Combine like terms: -6.666666667 + 6.666666667 = 0.000000000 0.000000000 + -2.666666667x + x2 = 0 + 6.666666667 -2.666666667x + x2 = 0 + 6.666666667 Combine like terms: 0 + 6.666666667 = 6.666666667 -2.666666667x + x2 = 6.666666667 The x term is -2.666666667x. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667x + 1.777777780 + x2 = 6.666666667 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667x + x2 = 6.666666667 + 1.777777780 Combine like terms: 6.666666667 + 1.777777780 = 8.444444447 1.777777780 + -2.666666667x + x2 = 8.444444447 Factor a perfect square on the left side: (x + -1.333333334)(x + -1.333333334) = 8.444444447 Calculate the square root of the right side: 2.905932629 Break this problem into two subproblems by setting (x + -1.333333334) equal to 2.905932629 and -2.905932629.Subproblem 1
x + -1.333333334 = 2.905932629 Simplifying x + -1.333333334 = 2.905932629 Reorder the terms: -1.333333334 + x = 2.905932629 Solving -1.333333334 + x = 2.905932629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 2.905932629 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 2.905932629 + 1.333333334 x = 2.905932629 + 1.333333334 Combine like terms: 2.905932629 + 1.333333334 = 4.239265963 x = 4.239265963 Simplifying x = 4.239265963Subproblem 2
x + -1.333333334 = -2.905932629 Simplifying x + -1.333333334 = -2.905932629 Reorder the terms: -1.333333334 + x = -2.905932629 Solving -1.333333334 + x = -2.905932629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -2.905932629 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -2.905932629 + 1.333333334 x = -2.905932629 + 1.333333334 Combine like terms: -2.905932629 + 1.333333334 = -1.572599295 x = -1.572599295 Simplifying x = -1.572599295Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.239265963, -1.572599295}Solution
x = {4.239265963, -1.572599295}
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